Global dynamics of the Buckingham’s two-body problem

© 2018, Springer Nature B.V. The equations of motion of the Buckingham system are the ones of a two-body problem defined by the Hamiltonian H=12(px2+py2)+Ae−Bx2+y2−M(x2+y2)3, where A, B and M are positive constants. The angular momentum pθ: = xpy− ypx and this Hamiltonian are two independent first integrals in involution. We denote by Ih (respectively, Ic) the set of points of the phase space where H (respectively, pθ) takes the value h (respectively, c). Due to the fact that H and pθ are first integrals, the sets Ih and Ihc= Ih∩ Ic are invariant under the flow of the Buckingham systems. We describe the global dynamics of the Buckingham system describing the foliation of its phase space by the invariant sets Ih, the foliation of the invariant set Ih by its invariant subsets Ihc, and the foliation of invariant set Ihc by the orbits of the system.